Percolation is Odd

Abstract

We prove a remarkable combinatorial symmetry in the number of spanning configurations in site percolation: for a large class of lattices, the number of spanning configurations with an odd or even number of occupied sites differs by $\pm 1$. In particular, this symmetry implies that the total number of spanning configurations is always odd, independent of the size or shape of the lattice. The class of lattices that share this symmetry includes the square lattice and the hypercubic lattice in any dimension, with a wide variety of boundary conditions.

arXiv comment4.5 pages, 1 figure

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Cited by in corpus (4)

• Bond percolation on simple cubic lattices with extended neighborhoods
• Exact Percolation Probability on the Square Lattice
• Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus
• On the oddness of percolation